Problem: What is the smallest whole number $b$ such that 62 can be expressed in base $b$ using only three digits?
Explanation: We are looking for the smallest base $b$ such that $100_b \le 62 < 1000_b$, which is the same as saying that $b^2 \le 62 < b^3$.  The smallest perfect cube greater than 62 is 64, so the smallest possible value of $b$ is $\sqrt[3]{64} = \boxed{4}$.